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A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
s and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s from 1 to . The
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .)


Formula

The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions n \times (n+1), which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: T_n = \frac . The example T_4 follows: This formula can be proven formally using
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
. It is clearly true for 1: T_1 = \sum_^k = \frac = \frac = 1. Now assume that, for some natural number m, T_m = \sum_^k = \frac. Adding m + 1 to this yields \begin \sum_^k + (m + 1) &= \frac + m + 1\\ &= \frac\\ &= \frac\\ &= \frac\\ &= \frac, \end so if the formula is true for m, it is true for m+1. Since it is clearly true for 1, it is therefore true for 2, 3, and ultimately all natural numbers n by induction. The German mathematician and scientist,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
, is said to have found this relationship in his early youth, by multiplying pairs of numbers in the sum by the values of each pair . However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans in the 5th century BC. The two formulas were described by the Irish monk Dicuil in about 816 in his Computus. An English translation of Dicuil's account is available. The triangular number solves the handshake problem of counting the number of handshakes if each person in a room with people shakes hands once with each person. In other words, the solution to the handshake problem of people is . The function is the additive analog of the factorial function, which is the ''products'' of integers from 1 to . The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation: L_n = 3 T_= 3;~~~L_n = L_ + 3(n-1), ~L_1 = 0. In the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, the ratio between the two numbers, dots and line segments is \lim_ \frac = \frac.


Relations to other figurate numbers

Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically, T_n + T_ = \left (\frac + \frac\right) + \left(\frac + \frac \right ) = \left (\frac + \frac\right) + \left(\frac - \frac \right ) = n^2 = (T_n - T_)^2. This fact can be demonstrated graphically by positioning the triangles in opposite directions to create a square: The double of a triangular number, as in the visual proof from the above section , is called a pronic number. There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula: S_ = 4S_n \left( 8S_n + 1\right) with S_1 = 1. ''All'' square triangular numbers are found from the recursion S_n = 34S_ - S_ + 2 with S_0 = 0 and S_1 = 1. Also, the square of the th triangular number is the same as the sum of the cubes of the integers 1 to . This can also be expressed as \sum_^n k^3 = \left(\sum_^n k \right)^2. The sum of the first triangular numbers is the th tetrahedral number: \sum_^n T_k = \sum_^n \frac = \frac . More generally, the difference between the th -gonal number and the th -gonal number is the th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number; the th centered -gonal number is obtained by the formula Ck_n = kT_+1 where is a triangular number. The positive difference of two triangular numbers is a trapezoidal number. The pattern found for triangular numbers \sum_^n_1=\binom and for tetrahedral numbers \sum_^\sum_^ n_1=\binom, which uses
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s, can be generalized. This leads to the formula: \sum_^\sum_^ \dots \sum_^\sum_^n_1 =\binom


Other properties

Triangular numbers correspond to the first-degree case of
Faulhaber's formula In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers :\sum_^n k^p = 1^p + 2^p + 3^p + \cdots + n^p as a (''p''&nb ...
. Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers. Every even
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
is triangular (as well as hexagonal), given by the formula M_p 2^ = \frac2 = T_ where is a
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...
. No odd perfect numbers are known; hence, all known perfect numbers are triangular. For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9: There is a more specific property to the triangular numbers that aren't divisible by 3; that is, they either have a remainder 1 or 10 when divided by 27. Those that are equal to 10 mod 27 are also equal to 10 mod 81. The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9". The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three. If is a triangular number, then is also a triangular number, given is an odd square and . Note that will always be a triangular number, because , which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for given is an odd square is the inverse of this operation. The first several pairs of this form (not counting ) are: , , , , , , ... etc. Given is equal to , these formulas yield , , , , and so on. The sum of the reciprocals of all the nonzero triangular numbers is \sum_^ = 2\sum_^ = 2 . This can be shown by using the basic sum of a telescoping series: \sum_^ = 1 . Two other formulas regarding triangular numbers are T_ = T_a + T_b + ab and T_ = T_aT_b + T_T_, both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra. In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including = 0), writing in his diary his famous words, " ΕΥΡΗΚΑ! ". This theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10 + 0), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the Fermat polygonal number theorem. The largest triangular number of the form is 4095 (see Ramanujan–Nagell equation). Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression. It was conjectured by Polish mathematician
Kazimierz Szymiczek Kazimierz (; la, Casimiria; yi, קוזמיר, Kuzimyr) is a historical district of Kraków and Kraków Old Town, Poland. From its inception in the 14th century to the early 19th century, Kazimierz was an independent city, a royal city of the Cr ...
to be impossible and was later proven by Fang and Chen in 2007. Formulas involving expressing an integer as the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function.


Applications

A fully connected network of computing devices requires the presence of cables or other connections; this is equivalent to the handshake problem mentioned above. In a tournament format that uses a round-robin group stage, the number of matches that need to be played between teams is equal to the triangular number . For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems. One way of calculating the depreciation of an asset is the sum-of-years' digits method, which involves finding , where is the length in years of the asset's useful life. Each year, the item loses , where is the item's beginning value (in units of currency), is its final salvage value, is the total number of years the item is usable, and the current year in the depreciation schedule. Under this method, an item with a usable life of = 4 years would lose of its "losable" value in the first year, in the second, in the third, and in the fourth, accumulating a total depreciation of (the whole) of the losable value.


Triangular roots and tests for triangular numbers

By analogy with the square root of , one can define the (positive) triangular root of as the number such that : n = \frac which follows immediately from the quadratic formula. So an integer is triangular
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
is a square. Equivalently, if the positive triangular root of is an integer, then is the th triangular number.


Alternative name

An alternative name proposed by
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer ...
, by analogy to factorials, is "termial", with the notation ? for the th triangular number.Donald E. Knuth (1997). ''The Art of Computer Programming: Volume 1: Fundamental Algorithms''. 3rd Ed. Addison Wesley Longman, U.S.A. p. 48. However, although some other sources use this name and notation, they are not in wide use.


See also

*
1 + 2 + 3 + 4 + ⋯ 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
*
Doubly triangular number In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if T_n=n(n+1)/2 denotes the nth triangular number, then the doubly triangul ...
, a triangular number whose position in the sequence of triangular numbers is also a triangular number * Tetractys, an arrangement of ten points in a triangle, important in Pythagoreanism


References


External links

*
Triangular numbers
at cut-the-knot
There exist triangular numbers that are also square
at cut-the-knot *
Hypertetrahedral Polytopic Roots
by Rob Hubbard, including the generalisation to ''triangular cube roots'', some higher dimensions, and some approximate formulas {{Classes of natural numbers Figurate numbers Factorial and binomial topics Integer sequences Proof without words Squares in number theory Triangles Simplex numbers